I want to get started casting bullets and loading my own bullets. I'm not sure if anyone here has experience with bullets, but I'm sure a few of you have experience with casting in general.

Basically, my questions are:
What kind of furnace do I need to melt down lead and copper? What do I need to do/wear to be safe? Where can I get good but inexpensive tools? Is there anything I should know first?

You don't need a furnace for lead, I've got a 60 year old electric melting pot.
Casting good bullets can actually be a bit tricky, to get good results you should pour in the first three or four rounds and wait a couple seconds before pulling the solid bullet out.
This gets the mould nice and hot, a cold mould doesn't produce very good bullets because the lead will cool down too quickly and leave wrinkles on the bullet which affects accuracy. You'll know when you're getting good bullets because they'll be smooth and shiny.
Likewise, don't drop them into a bucket of water, just set them down somewhere to air cool.

When I go to cast musket balls I have a candle burning and I hold the mould over the candle to blacken it, this makes it easy to remove the ball from the mould. Would do the same for any mould.

If you have a gun that fires balls, my favorite trick for after the balls are cast is to toss them in the back of the car or on an ATV in a metal ammo can. Doesn't take long before all the vibrations and bouncing around makes the balls perfectly smooth and round, even takes the sprue off.

You can get melting pots from places like Track of the Wolf or Cabela's. Just be sure you do it in a place with good ventilation, like outside. Beyond that you don't need any special safety equipment beyond some safety glasses, moulds have good long handles so just watch your hands.

You don't need a furnace for lead, I've got a 60 year old electric melting pot.
Casting good bullets can actually be a bit tricky, to get good results you should pour in the first three or four rounds and wait a couple seconds before pulling the solid bullet out.
This gets the mould nice and hot, a cold mould doesn't produce very good bullets because the lead will cool down too quickly and leave wrinkles on the bullet which affects accuracy. You'll know when you're getting good bullets because they'll be smooth and shiny.
Likewise, don't drop them into a bucket of water, just set them down somewhere to air cool.

When I go to cast musket balls I have a candle burning and I hold the mould over the candle to blacken it, this makes it easy to remove the ball from the mould. Would do the same for any mould.

If you have a gun that fires balls, my favorite trick for after the balls are cast is to toss them in the back of the car or on an ATV in a metal ammo can. Doesn't take long before all the vibrations and bouncing around makes the balls perfectly smooth and round, even takes the sprue off.

You can get melting pots from places like Track of the Wolf or Cabela's. Just be sure you do it in a place with good ventilation, like outside. Beyond that you don't need any special safety equipment beyond some safety glasses, moulds have good long handles so just watch your hands.

I've got both Lee and Lyman moulds and I like the Lee moulds more. Lyman has the occasional issue with quality, and they're around twice as expensive as Lee.

Lee moulds are made of aluminum as well so they won't rust like Lyman's steel moulds do (Which is a big bonus because you don't want your mould to have oil in it), you do need to be a bit more careful with them but if you make sure they're kept clean and handle them gently they'll last a lifetime. The newer Lee moulds tend to have more cavities than Lyman moulds too.

I'd be interested in this, too, but a fully lead bullet deposits shitloads of lead in the barrel of a gun when exiting the barrel at speeds of 300m/s or greater, and I doubt I could jacket bullets effectively.

Just curious, what are some useful things that can be created from a forge? I'm seriously considering undertaking the construction of a forge as a project over the summer, but I'd be more inclined to do so if I had some ideas of things to forge with it; other than filling up my garage with short-swords and cannonballs

If you're doing a blacksmithing forge, you can smith all sorts of things. Nails, lattice work, shovels, spades, any kind of tool really. Blacksmiths were immensely successful people, and some still are.

As far as metal-casting, still plenty of things. A lot of it will require further refining, usually with blacksmithing or other metal work. I've seen people build their own lathes and saws with cast metal, tools, grills, anything really.

If you're doing a blacksmithing forge, you can smith all sorts of things. Nails, lattice work, shovels, spades, any kind of tool really. Blacksmiths were immensely successful people, and some still are.

As far as metal-casting, still plenty of things. A lot of it will require further refining, usually with blacksmithing or other metal work. I've seen people build their own lathes and saws with cast metal, tools, grills, anything really.

I'm not really sure what the difference is. Couldn't a forge be built for blacksmithing and metal-casting? I'd probably be smithing things such as various tools, cutlery, and the occasional piece of whatever here and there. As for metal-casting, I'd probably be able to find something to do with that as-well. I'm just looking for a hobby that has at least some real-world application, and metal-working is something that has always kind of appealed to me, mostly because it is awesome.

No. A blacksmith forge doesn't need to get nearly as hot as a metal-casting forge. It doesn't take that much heat, relatively speaking, to get metal soft enough to beat it with a hammer. But to actually melt it, it takes VERY high temperatures. From what I've been looking up, forges for metal casting are often fueled with waste oil or propane sprayed into a firing chamber. The metal melting temperatures vary but for iron you need around 3600 degrees Fahrenheit or hotter.